Controlling Vortex Rotation in Dry Active Matter

TL;DR

This work addresses the problem of steering spontaneous vortex rotation in dry active matter around a central obstacle. By surrounding the obstacle with $M$ half-circles and rotating them by angle $\alpha$, the authors define the control parameter $\Pi_M = \frac{\langle \omega_M(t) \rangle}{\sqrt{\langle \omega_0^2(t) \rangle}}$ to compare the controlled vortex with the isolated one, and show two rotational regimes: $\Pi_M<0$ when flat sides face the vortex and $\Pi_M>0$ when curved sides face the vortex, corresponding to clockwise and counterclockwise rotation respectively. The study demonstrates that $M=4$ can yield the strongest, and even faster-than-isolated rotation ( $\Pi_M>1$ for small $\lambda$), while other configurations produce weaker effects; the rotation direction is controlled by $\alpha$ and vanishes near $\alpha=0$ or $\pm 90^{\circ}$. Additionally, the half-circles influence vortex stability in a manner dependent on obstacle size $D$ and gap $\lambda$, with a phase diagram revealing unstable, transient, and stable regimes and showing that large obstacles behave as effectively isolated vortices. Overall, the work provides a practical geometric route to control spontaneous vortex rotation in dry active matter and points to extending the scheme to obstacle lattices for guided lattice states.

Abstract

We investigate the rotation of a vortex around a circular obstacle in dry active matter in the presence of M half-circles distributed around the obstacle. To quantify this effect, we define the parameter ΠM , which is the ratio between the mean angular velocity of the controlled vortex and the root-mean-square angular velocity of the isolated vortex. We identify two rotational regimes determined by the obstacle configuration. In the first regime, where ΠM < 0 corresponding to the flat side of the half-circles facing the vortex, the rotation is clockwise. In the second regime (ΠM > 0), it corresponding to the curved sides facing the vortex, the rotation becomes counterclockwise. We further analyze the impact of this control on vortex stability, showing that the configuration of semi-circles can enhance or suppress stability depending on their geometry and distance from the central obstacle. Our results demonstrate a possible setup to control the spontaneous rotation of dry active matter around circular obstacles.

Figures (5)

• Figure 1: In the upper panels, two snapshots of the system configuration with obstacle size $D=20$ and $M=4$ half-circles, for: (a) $\alpha=50$, $\lambda=0.3D$ and (b) $\alpha=-50$, $\lambda=0.1D$. In lower panels, corresponding mean velocity fields averaged over 25 long-time realizations are presented in (c) and (d).
• Figure 2: $\Pi$ as a function of the angle $\alpha$ for different values of $\lambda$. Each panel shows the behavior of $\Pi$ for a distinct geometric configuration of half-circles surrounding the central obstacle, with the configuration displayed as an inset: (a) four half-circles (4HC), (b) three half-circles (3HC), (c) two adjacent half-circles (2AHC), (d) two opposite half-circles (2OHC), and (e) one half-circle (1HC). Panels (f)--(h) show snapshots of the system for an obstacle of size $D = 20$ with $M = 4$ half-circles at $\lambda = 0.2D$, for $\alpha = +90^\circ$, $0^\circ$, and $-90^\circ$, respectively.
• Figure 3: Normalized angular velocity probability distributions $P(\omega)$ for $\alpha = 50^\mathrm{o}$ in the 4HC for different $\lambda$ and different obstacle sizes: (a) $D = 10$, (b) $D = 20$, (c) $D = 30$, and (d) $D = 40$, and the dashed curves represent the isolated case, where box's size corresponds $L = 3D$.
• Figure 4: Phase diagram of $\lambda/D$ vs $D$ with $\alpha=50^{\mathrm{o}}$ and 4SD. Black circles, red squares and green diamonds represent the unstable, transient, and stable states, respectively.
• Figure 5: Dependence of $\Pi_{M}$ on $\lambda$ for $\alpha = 50^{\mathrm{o}}$ in the 4HC. Panels (a), (b) and (c) corresponds to the naturally random,transient and stable vortex states respectively.